I'm taking an introductory course on probability theory. We've been introduced to the function $P(E), E \subseteq S$ ($S$ being the sample space of some random process) which maps the event $E$ to its corresponding probability. Clearly, $P$ is a function from $\mathcal{P}(S)$ to $[0,1]$.
My question appears when we introduce conditional probability. By the above reasoning $P(A|B)\ A,B\subseteq S$ would imply that $A|B$ is a subset of $S$. My problem is that I cannot see how $A|B$ can be a subset of $S$. I don't even see how $A|B$ is a set, and even if it is a set I cannot visualize how it relates to $A$ and $B$. It is clearly distinct from $A\cap B$ since $P(A|B)=P(A\cap B)$ does not hold in general.
Have I misunderstood something? Is the definition of 'event' as a subset of the sample space incorrect? What is the domain of $P$?